Rather than attempting to parallel the proofs of Lemmas REF  and REF , one may attempt to mimic the proof technique of [1]} and prove the existence of subjective equilibrium using the properties of the subjective best-response correspondences. To use this approach, one may study the mappings
\(\mathbf {\pi }\mapsto {\rm Subj\text{-}BR}^i_{\epsilon } ( \mathbf {\pi }^{-i} , \mathcal {V}^{*}, \mathcal {W}^{*} ),\) 
